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\newcommand{\CourseName}{复变函数作业3ABC}
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\begin{document}

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\section{Elementary Point Set Topology }

\begin{enumerate}

\item  (***) %%1 
If $X$ is the set of complex numbers whose real and imaginary parts are rational, what is the interior $\mathrm{Int}\, X$, the closure $X^-$ and the boundary $\partial X$?

\item  (***) %%2
Let $E$ be the set of points $(x,y) \in \mathbb{R}^2$ such that $0\le x \le 1$ and either $y = 0$ or $y = 1/n$ for some positive integer $n$. 
\begin{enumerate}
\item[(a)]  What are the components of $E$? 
\item[(b)]  Are they all closed? 
\item[(c)]  Are they relatively open? 
\item[(d)]  Verify that $E$ is not locally connected.
\end{enumerate}

\item  %%3
Let $S$ be the set of all sequences $x = \{x_n\}$ of real numbers such that only a finite number of the $x_n$ are $\neq 0$. Define $d(x,y) =\max |x_n - y_n|$. Is the space complete? Show that the $\delta$-neighborhoods are not totally bounded.

\item  %%4
Which of the following functions are uniformly continuous on the whole real line:  $\sin x$, $x \sin x$, $x \sin (x^2)$, $|x|^{\frac{1}{2}} \sin x$?

\end{enumerate}

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\newpage
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\section{Conformality } 

\begin{enumerate}

\item  %%1
Let $\Omega$ be the complement of the negative real axis $z \le 0$; this set is open and connected. 
In $\Omega$ one and only one of the values of $\sqrt{z}$ has a positive real part.
With this choice $w = \sqrt{z}$ becomes a single-valued function in $\Omega$. 
\begin{enumerate}
\item  Prove that it is continuous.
\item  Prove that it is analytic. 
\end{enumerate}

\item  (***) %%2
Let $w=f(z)$ be an analytic function in the region $\Omega$. Let $z_0\in \Omega$. 
\begin{enumerate}
\item  Prove that two curves which form an angle at $z_0$ are mapped upon curves forming the same angle, in sense as well as in size. 
\item  Prove that the linear change of scale at $z_0$, effected by the transformation $w = f(z)$, is independent of the direction. 
\end{enumerate}

\item  %%3
Let $w=f(z)$ be an analytic function in the region $\Omega$. 
Let $z_0\in \Omega$ and assume that $f'(z_0)\neq 0$. 
Prove that the mapping is topological (i.e., a homeomorphism) if it is restricted to a sufficiently small neighborhood of $z_0$. 

\item (***) %%4
Let $w=f(z)$ be an analytic function in the region $\Omega$. 
\begin{enumerate}
\item  Let $\gamma\subset \Omega$ be a differentiable arc with the equation $z=z(t), a\le t\le b$. Prove that the length of the image $\gamma'=f(\gamma)$ is given by 
\begin{equation*}
L(\gamma') = \int_{\gamma} |f'(z)||dz|. 
\end{equation*}
\item  Let $E\subset \Omega$ be a point set. Prove that the area of the image $E' = f(E)$ is given by 
\begin{equation*}
A(E') = \iint_E |f'(z)|^2dxdy.
\end{equation*}
\end{enumerate}

\end{enumerate}

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\newpage
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\section{Linear Transformations } 

\begin{enumerate}

\item  %%1
Explain that the linear transformation $$w=\frac{az+b}{cz+d}$$ is a topological mapping of the extended plane onto itself, the topology being defined by distances on the Riemann sphere.

\item (***)  %%2
Explain that the linear transformation 
\begin{equation*}
w=\frac{az+b}{cz+d}
\end{equation*}
is composed by a translation, an inversion, a rotation, and a homothetic transformation followed by another translation. 

\item (***)  %%3
Prove that the cross ratio $(z_1,z_2,z_3,z_4)$ is real if and only if the four points lie on a circle or on a straight line.

\item (***)  %%4
Find the linear transformation which carries $0, i, -i$ into $1, -1, 0$.

\item  %%5
Explain the geometric construction of the symmetric point of $z$. The symmetry is with respect to the circle $C$ of center $a$ and radius $R$. 

\item  %%6
Prove the symmetry principle. If a linear transformation carries a circle $C_1$ into a circle $C_2$, then it transforms any pair of symmetric points with respect to $C_1$ into a pair of symmetric points with respect to $C_2$.

\item (***)  %%7
Find a linear transformation which carries $|z| = 1$ and $|z - \frac{1}{4}| = \frac{1}{4}$ into concentric circles. What is the ratio of the radii?

\item  %%8
If $z_1, z_2, z_3, z_4$ are points on a circle, show that $z_1, z_3, z_4$ and $z_2, z_3, z_4$ determine the same orientation if and only if $(z_1,z_2,z_3,z_4) > 0$.

\end{enumerate}

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\end{document}

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